Conformal capacity of hedgehogs

Abstract

We discuss problems concerning the conformal condenser capacity of “hedgehogs”, which are compact sets E E in the unit disk D = { z : | z | > 1 } \mathbb {D}=\{z:\,|z|>1\} consisting of a central body E 0 E_0 that is typically a smaller disk D ¯ r = { z : | z | ≤ r } \overline {\mathbb {D}}_r=\{z:\,|z|\le r\} , 0 > r > 1 0>r>1 , and several spikes E k E_k that are compact sets lying on radial intervals I ( α k ) = { t e i α k : 0 ≤ t > 1 } I(\alpha _k)=\{te^{i\alpha _k}:\,0\le t>1\} . The main questions we are concerned with are the following: (1) How does the conformal capacity c a p ( E ) \mathrm {cap}(E) of E = ∪ k = 0 n E k E=\cup _{k=0}^n E_k behave when the spikes E k E_k , k = 1 k=1 , …, n n , move along the intervals I ( α k ) I(\alpha _k) toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity c a p ( E ) \mathrm {cap}(E) depend on the distribution of angles between the spikes E k E_k ? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, will also be suggested.